Question

simplify.

1. ((n^4)^{\frac{3}{2}})
2. ((a^8)^{\frac{3}{2}})
3. ((64m^4)^{\frac{3}{2}})
4. ((81p^4)^{\frac{3}{2}})
5. ((49n^2)^{\frac{1}{2}})
6. ((4k^4)^{\frac{3}{2}})
7. ((64x^6)^{\frac{3}{2}}) (with some handwritten notes and calculations)
8. ((9r^4)^{0.5}) (converted to ((9r^4)^{\frac{1}{2}}) with handwritten work)
9. ((125n^6)^{\frac{5}{3}})
10. ((27p^6)^{\frac{5}{3}}) (with handwritten calculations including calculator input notes)
11. ((216r^9)^{\frac{1}{3}}) (with handwritten cube root work)
12. ((16x^{12})^{\frac{5}{4}})
13. ((81v^8)^{\frac{3}{4}})
14. ((x^{16})^{\frac{5}{4}})
15. (3b^{\frac{1}{2}} cdot b^{\frac{4}{3}})

Explanation:

Step1: Apply power rule $(x^a)^b=x^{a \cdot b}$

For each term, multiply exponents:

1. $(n^4)^{\frac{3}{2}} = n^{4 \cdot \frac{3}{2}} = n^6$
2. $(a^8)^{\frac{3}{2}} = a^{8 \cdot \frac{3}{2}} = a^{12}$
3. $(64m^4)^{\frac{3}{2}} = 64^{\frac{3}{2}} \cdot m^{4 \cdot \frac{3}{2}} = (8^2)^{\frac{3}{2}} \cdot m^6 = 8^3 \cdot m^6 = 512m^6$
4. $(81p^4)^{\frac{3}{2}} = 81^{\frac{3}{2}} \cdot p^{4 \cdot \frac{3}{2}} = (9^2)^{\frac{3}{2}} \cdot p^6 = 9^3 \cdot p^6 = 729p^6$
5. $(49n^2)^{\frac{1}{2}} = 49^{\frac{1}{2}} \cdot n^{2 \cdot \frac{1}{2}} = 7n$
6. $(4k^4)^{\frac{3}{2}} = 4^{\frac{3}{2}} \cdot k^{4 \cdot \frac{3}{2}} = (2^2)^{\frac{3}{2}} \cdot k^6 = 2^3 \cdot k^6 = 8k^6$
7. $(64x^6)^{\frac{3}{2}} = 64^{\frac{3}{2}} \cdot x^{6 \cdot \frac{3}{2}} = (8^2)^{\frac{3}{2}} \cdot x^9 = 8^3 \cdot x^9 = 512x^9$
8. $(9r^{-4})^{0.5} = (9r^{-4})^{\frac{1}{2}} = 9^{\frac{1}{2}} \cdot r^{-4 \cdot \frac{1}{2}} = 3r^{-2} = \frac{3}{r^2}$
9. $(125n^6)^{\frac{5}{3}} = 125^{\frac{5}{3}} \cdot n^{6 \cdot \frac{5}{3}} = (5^3)^{\frac{5}{3}} \cdot n^{10} = 5^5 \cdot n^{10} = 3125n^{10}$
10. $(27p^6)^{\frac{5}{3}} = 27^{\frac{5}{3}} \cdot p^{6 \cdot \frac{5}{3}} = (3^3)^{\frac{5}{3}} \cdot p^{10} = 3^5 \cdot p^{10} = 243p^{10}$
11. $(216r^9)^{\frac{1}{3}} = 216^{\frac{1}{3}} \cdot r^{9 \cdot \frac{1}{3}} = (6^3)^{\frac{1}{3}} \cdot r^3 = 6r^3$
12. $(16x^{12})^{\frac{5}{4}} = 16^{\frac{5}{4}} \cdot x^{12 \cdot \frac{5}{4}} = (2^4)^{\frac{5}{4}} \cdot x^{15} = 2^5 \cdot x^{15} = 32x^{15}$
13. $(81v^8)^{\frac{3}{4}} = 81^{\frac{3}{4}} \cdot v^{8 \cdot \frac{3}{4}} = (3^4)^{\frac{3}{4}} \cdot v^6 = 3^3 \cdot v^6 = 27v^6$
14. $(x^{16})^{\frac{5}{4}} = x^{16 \cdot \frac{5}{4}} = x^{20}$
15. Apply product rule $x^a \cdot x^b = x^{a+b}$:

$3b^{\frac{1}{2}} \cdot b^{\frac{4}{3}} = 3b^{\frac{1}{2}+\frac{4}{3}} = 3b^{\frac{3+8}{6}} = 3b^{\frac{11}{6}}$

Answer:

1. $n^6$
2. $a^{12}$
3. $512m^6$
4. $729p^6$
5. $7n$
6. $8k^6$
7. $512x^9$
8. $\frac{3}{r^2}$
9. $3125n^{10}$
10. $243p^{10}$
11. $6r^3$
12. $32x^{15}$
13. $27v^6$
14. $x^{20}$
15. $3b^{\frac{11}{6}}$